Summing over a range distributes over pointwise addition of rational sequences

Identifier: F:rat-sumrange-add
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For rational sequences f and g and a bound n, sum_{i<n} (f i + g i) = sum_{i<n} f i + sum_{i<n} g i.

Formal statement

theorem Rat.sumRange_add : ((x0 : ((x0 : AxNat) -> Rat)) -> ((x1 : ((x1 : AxNat) -> Rat)) -> ((x2 : AxNat) -> Eq.{1} Rat (Rat.sumRange (fun (x3 : AxNat) => Rat.add (x0 x3) (x1 x3)) x2) (Rat.add (Rat.sumRange x0 x2) (Rat.sumRange x1 x2)))))

Dependencies

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2 direct dependencies 6 direct dependents

Evidence

kernel-Rat.sumRange_add

Kind: kernel-term
Status: checked

Supports: Rat.sumRange_add is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Rat.sumRange_add 2>/dev/null | grep -cE '^Rat\.sumRange_add[[:space:]]'

Evidence notes

`build_rat_prelude` admits Rat.sumRange_add through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so a line in this tool's output for the exact name is a machine-checked proof having been admitted. `theorem_dependency_inventory` exits non-zero for a named filter that matches nothing (a deleted theorem cannot read as a re-derived one), and `grep -c` (never `-q`) both avoids a SIGPIPE-under-pipefail false negative and independently asserts the exact tab-anchored line is present -- two ways the same absence would be caught. `--release` is MANDATORY: this tool now also builds `creal`/`complex`/`cpoint`, which recurse deep enough in a debug build to overflow the default thread stack (measured: release exits 0 with 1092 theorems, debug SIGABRTs at 134) -- the same resource-limit gotcha already documented for `prelude_theorem_inventory --include-constructed`.

footprint-Rat.sumRange_add

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Rat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free rat

Evidence notes

`nat_axiom_inventory` builds the Rat environment as its own group and reports `rat: axiom=0 opaque=0 quotient=0 total_trusted=0` (re-measured on this tree). That bounds every individual Rat theorem's footprint by [], since a theorem cannot depend on a trusted declaration the environment does not contain -- and the enumeration covers Axiom, Opaque AND Quotient, not just Declaration::Axiom, because Opaque has no proof body and Quotient admits Quot.sound. `--require-axiom-free <name>` is an error (not a silent pass/zero) for a prelude this run never built, which is what makes `rat` here a claim rather than an absence.

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_rat_prelude",
  "source": "see F:rat-add-comm for the extraction method (shared for the whole batch)."
}